Avoid overflow in gamma_inc derivatives

[vera-rubin]

Differentiating gamma_inc(a, x, 0) with respect to x at a = x = 150.0 or 200.0 returns Inf or NaN:

using ForwardDiff, SpecialFunctions
for a in (150.0, 200.0)
    @show ForwardDiff.derivative(x -> gamma_inc(a, x, 0)[1], a)
end

The documented normalization and derivative identity give finite derivatives of about 0.03255540946 and 0.02819772769, respectively.

We evaluate the derivative in log space using loggamma to avoid overflowing intermediate values while retaining the power rule at zero for higher derivatives.

The new regression tests and the existing dual-number tests pass.

Tested versions: ForwardDiff 1.4.6, Julia 1.10.12, SpecialFunctions 2.9.0.

— ChipLab

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 91.25%. Comparing base (a3c0f4f) to head (a90c217).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #862      +/-   ##
==========================================
+ Coverage   91.23%   91.25%   +0.02%     
==========================================
  Files          11       11              
  Lines        1072     1075       +3     
==========================================
+ Hits          978      981       +3     
  Misses         94       94              

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